The Pinhole Camera

An ideal model of a camera is the pinhole camera, as seen in Figure 2.2. This kind of camera can be imagined as a box with a pinhole, through which light enters and forms a two-dimensional image on the opposite site. A point in the three-dimensional -space is projected to an image-point in the two-dimensional -space (wall). If the coordinate system of the-space is aligned at the pinhole so that the Z-axis coincides with the optical axis and the image plane has its origin at , then the projection equations are given by

(2.4)

and

(2.5)

Figure 2.2: Pinhole camera model

To represent Equation 2.4 and Equation 2.5 in a linear way, we transform the point in the Euclidean plane to a point in the projective plane. This represents the same point, we simply added a new coordinate . Overall scaling is unimportant. The point can be re-transformed by dividing through . Thus is similar to . Because scaling is unimportant, the coordinate is called homogeneous coordinate. Homogenous coordinates can also be used within a higher dimensional domain. Now we can combine Equation 2.4 and Equation 2.5 to

(2.6)

If we want to know the image coordinates we have to take four more values into account. Namely

image principal point, which is the intersection between the camera’s optical axis and the image plane.

distance between two sensor elements in and direction

and can normally be found in the datasheet2.2 of the sensor chip. The image principal point has to be found by calibration (see Section 2.2 for more information). If the parameters and the point in camera coordinates are known, we can compute the image coordinates with the following formulation

(2.7)

and

(2.8)

where and are the coordinates of the point in the camera coordinate system. If we combine Equation 2.6 with Equation 2.7 and Equation 2.8 we can formulate the translation from Point to the image coordinates